Properties Of Triangles When They Undergo The Curve-Shortening Flow

Transcription

1 Properties Of Triangles When They Undergo The Curve-Shortening Flow Asha Ramanujam under the direction of Mr Ao Sun Department of Mathematics Massachusetts Institute of Technology Research Science Institute August, 016

2 Abstract In this paper, we study the properties of triangles when they undergo the curve-shortening flow We first take the case of the isosceles triangle and prove that the legs of the triangle always decrease and the vertex angle will decrease if it is lesser than Our main result is 3 that there exists a time T such that αt = 0 where αt is the vertex angle at time t and xt > 0 where xt is the legs of isosceles triangle at time t when 0 α0 <, 3 showing that the triangle becomes a straight line rather than a single point The equilaterel triangle turns out to be a self-shrinker and hence converges to a single point We also find expressions for the change in side lengths and angles in any general triangle and prove that the length of each side reduces during the flow Summary Geometric flow refers to the movement of geometric objects in space The curve-shortening flow is an important type of flow in which all points of the curve move inwards It is highly applied in physics and recent research in this field addresses classical problems in topology, differential equations and geometric analysis Our aim in this paper, is to study triangles when they undergo this flow We find that the length of the legs of an isosceles triangle decreases We also find that the vertex angle of an isosceles triangle decreases if it is smaller than and ultimately, the isosceles triangle becomes a line segment We find that the equilaterel triangle undergoes the flow without having any change in its angles, and eventually 3 converges to a point We also find that in any triangle, the length of each side decreases during the flow, and we can roughly estimate what happens to the angles of the triangle

3 1 Introduction In this project we study the curve-shortening flow CSF, a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature at this point This process is of fundamental interest in the calculus of variations Recent research on curve-shortening flow addresses classical problems in topology, differential equations and geometric analysis Furthermore it provides a setting for formulating problems in physics It covers concepts like the area form and total curvature The curve-shortening flow was originally studied as a model for the annealing of metal sheets Later, it was applied in image analysis to give a multi-scale representation of shapes It can also model reaction diffusion systems, and the behavior of cellular automata The curve-shortening flow can be used to find closed geodesics on Riemannian manifolds, and as a model for the behavior of higher-dimensional flows [1] The curve-shortening flow follows the equation dx i t = kx i t Nx i t where x i t is a point on the curve on the curve moving at time t, Nx i t is the normal of the curve at the point x i t and kx i t is the curvature of the curve at the point x i t The Gage-Hamilton-Grayson theorem states how smooth curves evolve under the curveshortening flow [] It states that, if a smooth simple closed curve undergoes the curveshortening flow, it remains smoothly embedded without self-intersections It will eventually becomes convex, and once it does so, it will remain convex After this time, all points of the curve will move inwards, and the shape of the curve will converge to a circle as the whole curve shrinks to a single point Figure 1 This behavior is sometimes summarized by saying that every smooth simple closed curve shrinks to a round point Gage proved the convergence to a circle for convex curves that 1

4 Figure 1: Evolution of triangle under CSF contract to a point Gage and Hamilton proved that all smooth convex curves eventually contract to a point, and Grayson proved that every non-convex curve will eventually become convex [1] In this paper, we will study how the triangles evolve under the curve-shortening flow In Section we will present some definitions and notations involved through out the paper We investigate the behaviour of the legs and the vertex angle of an isosceles triangle under the curve-shortening flow in Section 3, and eventually what happens to it In Section we will make some calculations for any triangle and formulate some observations based on some graphical evidence We find that the length of the legs of an isosceles triangle always decreases We also find that the vertex angle decreases if it is lesser that 3 and remains the same if it is 3 Surprisingly, we show that unlike smooth curves, in particular for isosceles triangles, if the vertex angle is lesser that, the triangle ultimately becomes a short line segment and not 3 a single point We also find that in any triangle the length of each side reduces during the curve-shortening flow In the end we come up with some conjectures, based on some computation and graphical evidence Notations and Definitions Let there be a triangle ABC which will undergo the curve-shortening flow Figure We will use the following notations in the paper

5 Figure : an arbitary triangle at time t At any time t, A t, B t and C t denote the flowed vertices of A, B and C, respectively The lengths of the sides A t B t and B t C t are denoted by x t and p t, respectively at any time t For convinience, we assume C 0 to be the origin and B 0 C 0 to be the x-axis of the Cartesian coordinate system The angles at the flowed vertices at any time t, are denoted by αt, βt and γt respectively Definition 1 Curvature at any vertex is defined as angle at that vertex Definition The normal at any vertex is defined as its angle bisector Definition 3 Self shrinker is a figure which undergoes the curve-shortening flow without changing its angles In other words α t = α 0, β t = β 0, γ t = γ 0, for any time t 3 Isosceles triangle Let us consider an isosceles triangle ABC with AB = AC According to our assumptions, 3

6 A 0 : B 0 : x 0 sin x 0 sin α0 α0, x 0 cos, 0 α0, When it undergoes the curve-shortening flow in an infinitesimally small time t the following transformation Figure 3 takes place: A t: x 0 sin α0, x 0 cos α0 α 0 t, C t: + α0 cos α0 t, + α0 sin B t: x 0 sin α0 + α0 cos α0 t, α0 t + α0, sin α0 t Figure 3: Isosceles triangle undergoing the curve-shortening flow Next we prove 3 theorems about the behavior of the legs and the vertex angle during the curve-shortening flow Theorem The length of the legs of an isosceles triangle, decreases during the curveshortening flow Proof We find dxt to prove our theorem

7 We find it in the following waydx t t=0 = x 0 dx t x t x 0 t=0 = lim t 0 t Therefore, = x 0 + α 0 sin + α 0 + α 0 cos α 0 dx t t=0 = + α 0 sin + α 0 + α 0 cos α 0 By the same method, dx t = + α t sin + α t α t + α t cos Since the expression + αt sin + αt + α t cos αt is negative for all α t, such that 0 < α t <, dxt is negative implying that x t always decreases We find dαt for further calculations To find dαt, we proceed as follows d cos α t t=0 = sin α t dα t cos α t cos α 0 t=0 = lim t 0 t 5

8 = sin α 0 x 0 + α 0 1 sin α0 α 0 α 0 sin Therefore, dα t t=0 = x 0 + α 0 1 sin α0 α 0 α 0 sin By the same method dα t = x t + α t 1 sin αt α t α t sin For convinience we set fs to be, Therefore, f s = + s 1 sin s s s sin fs = 1 s s + s cos + s sin s s sin We use the following lemma for further calculations Lemma 1 The function f s is positive for 0 s < 3 Proof We show that dfs ds that interval < 0 to prove that fs is a monotonically decreasing function in 6

9 df s ds = 1 s s cos s + s + sin s s cos + s sin Case 1 0 s < In this interval cos s > sin s So, df s ds < 1 s s cos s sin + s + s s cos Since cos s 1, and cos s > cos 8 09, df s ds < 1 s s 09 < 0 Case s = dfs ds 0836 < 0 Case 3 < s 3 In this interval, cos s > sin s So, df s ds < 1 s s cos s sin + s + s s cos Since cos s < 1, sin s > sin and cos s 3, df s ds < 1 s + + s 0195 s 3 < 0 Therefore dfs ds < 0 for 0 s 3 function in that interval showing that f s is a monotonically decreasing Furthermore, f 3 = 0 and f 0 > 0, showing that f s > 0 for 0 s < 3 7

10 Based on the previous computations of Lemma 1, dαt = 0 at αt =, leading to our Theorem 5 3 dαt < 0 for 0 αt < 3 Also Theorem 5 The vertex angle αt of an isosceles triangle undergoing the curve-shortening flow, decreases if αt < 3 and remains same if αt = 3 Now we will analyse what will happen to the triangle if α0 < 3 Theorem 6 The vertex angle vanishes in a shorter time than than the legs of an isosceles triangle undergoing the curve-shortening flow, if α0 <, ie there exists a time T such 3 that αt = 0 and xt > 0 Proof As we found in the previous part, dx t = + αt sin + α t α t + α t cos and dα t = x t + α t 1 sin αt α t α t sin Let the function gαt be defined as follows: g α t = + α t sin + α t + α t cos α t The function g α t satisfies g α t < for all t, since αt 3 This was found by substituting αt = 0, with g0 = + < Therefore, there exists a positive integer C such that, g α t < C and C Also, by the proof of Lemma 1, f α t f α 0 = C 1, where C 1 is a positive integer For any time t in the time interval [0, T, x t > 0 8

11 By Lemma 1, f α t > 0 and g α t > 0 as it is bounded between positive real numbers The functions α t and xt are decreasing in this interval Let t 0 = 0 and t i+1 = t i + xt i C Then for any positve integer n, by Lagrange Mean Value Theorem, there exists time t in the interval t n, t n+1 such that Therefore, x t n+1 x t n = t n+1 t n dx t x t n+1 = x t n t n+1 t n g α t Since g α t < C, we have x t n+1 > x t n t n+1 t n C Hence, x t n+1 > xtn By induction from 0 to n, x t n > xt 0 n Thus, x t is positive for all t in the interval [0, t n ], for all positive integers n By applying Lagrange Mean Value Theorem again, for any positive integer n, there exists a time s in the interval t n, t n+1 such that Therefore, α t n+1 = α t n + t n+1 t n dα s α t n+1 = α t n t n+1 t n f α s x s 9

12 Since f α s > C 1, we have α t n+1 < α t n C 1x t n C x s Since x t is a decreasing function, x t n > x s Therefore, x t n+1 < x t n C 1 C By induction, α t n < α t 0 n C 1 C Suppose there is a positive integer N such that N C 1 C > α t 0 Then we know that α t vanishes in the interval [0, t n ] because αt 0, but x t is positive in this interval The theorem implies that the triangle will become a straight line segment and not a single point after some timefigure Figure : the end of the CSF General Triangle Let us consider the triangle ABC Recall that pt denotes the side B t C t 10

13 According to our assumption, A 0 : p sin β 0 cos γ 0, sin β 0 + γ 0 B 0 : p 0, 0 p sin β 0 sin γ 0 sin β 0 + γ 0 When the triangle undergoes the curve-shortening flow in an infinitesimally small time t, the following transformation Figure 5 takes place The coordinates of the new vertices are - p0 sinβ0 cosγ0 A t = β0 + γ0 sin β0 sinβ0+γ0 γ0 cos β0 γ0 t B t = p 0 β 0 cos C t = γ 0 cos γ0 β0 γ0 t, β 0 sin t, γ 0 sin γ0 t, p0 sinβ0 sinγ0 sinβ0+γ0 β0 + β0 t t Figure 5: A general triangle undergoing the CSF Next we prove 1 theorem and verify conjectures Theorem 7 In any triangle, the length of each side of the triangle, decreases during the 11

14 curve-shortening flow Proof We find dpt to prove our theorem We find it in the following way Therefore, dp t dp t p t p 0 t=0 = p 0 t=0 = lim t 0 t β 0 γ 0 = p0 β 0 cos + γ 0 cos dp t β 0 γ 0 t=0 = β 0 cos + γ 0 cos By the same method, dp t β t γ t = β t cos + γ t cos Since the expression β t cos β t < and 0 < γ t <, dpt βt + γ t cos γt is positve for all 0 < is negative proving that the length of side BC decreases The same computation can be done for all sides and a similar result will be obtained which is based on the adjacent angles, and the derivative of the length of the side with respect to time being negative 1 Other Results and Conjectures We find that: dβt cos 3βt = cscγt pt βt cos βt γt βt cos3 βt + γt γt sin γt βt sinγt + βt + γtsin γt 3γt +γt+cos cos + + γt + sin 3βt+γt 1

15 dγt = cscβt pt βt cosβt+βt cos3 βt γt +γt sinβt+ sinγt sinβt + γt βt sinγt + βt + γtsin + γt + sin 3βt+γt Conjecture 8 In any triangle which is not equilateral, the smallest angle decreases during the curve-shortening flow If we assume in the triangle that γ 0 α 0 β 0, then the following 3D graph Figure 6 depicts dβt p t Figure 6: graph of dβt p t The graph shows that dβt p t is always negative, implying that dβt is negative, showing that β decreases and verifying our Conjeucture 8 Conjecture 9 The equilateral triangle is the only triangle which is a self shrinker If we apply α 0 = γ 0 = β 0 = dβt, then we get 3 = dγt = dαt = 0, proving that the equilateral triangle is a self shrinker Based on extensive computations, we can conjecture that it is the only self shrinker 5 Conclusion In this paper we examined some of properties triangles undergoing the curve-shortening flow In the case of the isosceles triangle, we found that the length of its legs decreases and its 13

16 vertex angle decreases if it is smaller than and remains the same at Furthermore, the 3 3 vertex angle in an isosceles triangle vanishes in a shorter time than the length, if the angle is smaller than 3 and the triangle eventually becomes a line segment and not a point In any triangle, the length of each side decreass during the flow Our future work is to study the behaviour of isosceles triangles when the vertex angle is greater than 3 and complete the observations by closely studying the behavior of any random triangle under the curveshortening flow 6 Acknowledgments I would like to thank my mentor Ao Sun of MIT for guiding me through this project and sharing with me inspirational ideas for exploration I would like to thank Dr Tanya Khovanova, Slava Gerovitch, Professor Ankur Moitra and Professor David Jerison for helping me in every possible way in this project I would also like to thank my tutor Dr Jenny Sendova for her insightful discussion in every step of the project I would like to thank Professor Skinner for encouraging me in this project I would like to thank Stanislav Atanasov and Sanath K Devalapurkar for helping me write the paper I would like to thank the TA s and nobodies I would also like to thank Sam Backwell, DrRickert, Jan Olivetti and Pawel Burzynski for helping me in the technical details I would like to thank the MIT math Department Last but not the least, I would like to thank CEE, RSI, my sponsors and my school for helping me get into the program 1

17 References [1] Curve shotening flow Available at [] SSmith, EBroucke, and BFrancis Curve shortening and the rendezvous problem for mobile autonomous robots Available at 15